Expectations - Ms. Hogan's Classes · 2018. 2. 26. · How To Win: If a player chooses a man followed by a Muppet they win $40. If they choose two Muppets they will win $5. If they

© 21st Century Math Projects

Expectations

How do you make decisions? Some will opt for a random strategy, some will

trust their gut and others will try to make decisions that make the most sense.

One such mathematical strategy to analyze these types of decisions is expected value. Expected value is the

most likely value that would occur if the situation happened infinitely many times. By understanding

expected value, people can make stronger decisions and make numerical sense to a variety of problems.

In each of the following problems, create probability distribution tables, calculate the expected value and

make a decision based on the data.

-Raffle Time-

Raffles are a popular way to raise money for different causes. Often a reward is given to attract people to buy

the tickets. Two different raffles are going on, which one gives a buyer the best chance to win?

The Raffle Queen

Snappy Raffle

1000 raffle tickets are sold for $3.00 each. There is

one grand prize for $750 and two consolation prizes

of $200 each that will be awarded. What is the

expected value of one ticket?

Outcome

Probability

250 raffle tickets are sold for $1.00 each. There is

one grand prize for $150 and three consolation

prizes of $25 each that will be awarded. What is the

expected value of one ticket?

Outcome

Probability

Which raffle has the higher expected value? If you are going to buy a ticket for one of the raffles, which

one would you pick?


-Hack a Dwight?-

During the early 2000’s a strategy to defend NBA legend Shaquille O’Neal was to intentionally foul him as soon

as he got the basketball. Teams felt that O’Neal, a historically bad foul shooter, would score less points if he took

foul shots than if he took a regular shot in the game. This strategy became known as “Hack a Shaq”. In the early

2010’s a similar strategy has at times been used against star center Dwight Howard. Is this a good strategy? How

does Howard’s expected values compare to O’Neal’s?

Shaquille O’Neal

Dwight Howard

In his career O’Neal made 58.2% of his field goal

attempts during the game and 52.7% of his free

throws.

Field Goal Attempts (taking a shot)

Outcome 2 0

Probability

Shooting 2 Free Throws

Outcome 2 1 0

Probability

In his career Howard has made 57.7% of his field

goal attempts during the game and 57.6% of his free

throws.

Field Goal Attempts (taking a shot)

Shooting 2 Free Throws

Is “Hack-a-Shaq” an effective strategy for either O’Neal or Howard? Why or why not?

If a team debates whether they should intentionally foul Dwight Howard, what would you suggest?

(Make 1st AND 2nd) (Miss 1st AND 2nd) (NOT 2 OR 0)

(Same idea as with Shaquille O'Neal data)

(Same idea as with Shaquille O'Neal data)

points points

points point points


-SAT Time-

On some standardized tests like the SAT, there is a penalty for getting a question wrong. On the SAT, a test taker

earns 1 point for every correct answer and subtracts ¼ for each incorrect answer.

A test taker has a question blank and less than a ten seconds to go in the test. The question has 5 multiple

choice options, should the test taker guess or not?

If the test taker can eliminate an answer choice, how much does it improve their odds?

Outcome 1 -1/4

Probability .2 .8

Outcome 1 -1/4

Probability .25 .75

-Health Insurance Premiums-

In most businesses, companies are given health insurance premiums (how much it cost for each employee) from

insurance companies based on the amount of usage from that business for the previous year. In this example,

determine the appropriate insurance premium for this company.

An insurance agency has compiled data for a company into the following chart, how much should they

charge as an annual premium for each employee in order to cover the costs?

Last year the company paid a $525 monthly premium for their

employees. They cannot afford any health insurance cost increases and

will need to pass along any additional costs to its employees. How

much will the employees need to pay?

Amount of

Annual Claims Probability

<4000 .35

6000 .28

8000 .15

10000 .12

>12000 .10


-Investment Portfolio-

When selecting a retirement account or a college savings account, there are a number of options with how to

partition the money. Whenever an investment is made there is a degree of risk involved. With $50,000 to invest,

consider the three investment options below and choose the best one.

Conservative

Aggressive

High Risk

Outcome Probability

25,000 .01

40,000 .22

50,000 .46

60,000 .28

100,000 .03

Outcome Probability

25,000 .06

40,000 .23

50,000 .25

60,000 .38

100,000 .08

Outcome Probability

25,000 .16

40,000 .31

50,000 .12

60,000 .26

100,000 .15

Which plan should will you invest in? Why?


The Inexpensive &

Odd Game Room

The first rule of the Inexpensive & Odd Game Room is that you don’t talk about the Inexpensive & Odd Game

Room. For this occasion game makers were invited to create their strangest games. And boy, did they deliver.

In this assignment, you will analyze the games available to play and determine the expected value of each to

decide which give you the best chances at the desired outcome!

Game Play:

For $1 a player has a chance to win $10. First, the player will choose one of

three rocks (igneous, sedimentary and metamorphic). Second, they will roll

a 6-sided die.

Game Pieces: 3 rocks and 1 die.

How To Win: If a player chooses the igneous rock and rolls a 6 then they will win $10. If

they do not chose the igneous rock, but roll a 6 then they will win $1.

Number of

Possible Outcomes

Probability

Distribution

+9 0 -1

Expected Value

Outcome ($)

Probability

+$9 $0 -$1

(Win $10 - cost of $1 to play) (Win $1 - cost of $1 to play) (Win Nothing - cost of $1 to play)


Game Play:

For $3 a player has a chance to win $250. First, the player will spin a spinner

with 4 equal regions denoted by suits. Second, they will randomly select a

card from a standard 52-card deck.

Game Pieces:

and a standard 52-card deck with all four suits.

How To Win:

A player must spin and land on the heart sector. Then they must select the

Jack, Queen or King hearts out of a random 52 card deck to win $250. If they

select any Ace, then they can they will win $5.

Number of Possible

Outcomes

Probability

Distribution

Expected Value

Game Play:

For $10 a player has a chance to win $30 if they can choose a card with a

marsupial on it. A player will draw a random animal card. The problem is

all the animals are from Oceania.

Game Pieces:

Short-nosed Echidna Cuscus Quoll Kiwi

Tasmanian Devil Goanna Komodo Dragon Platypus

Tawny Frogmouth Tarsier Dingo Dugong

12 Cards

How To Win: If the card is a bird, the player wins $5. If the player chooses a marsupial,

they win $30. All other cards lose.

Number of

Possible Outcomes

Probability

Distribution

Expected Value

and the spinner lands on the heart sector, they will win $5.

**Hint: Marsupials: Tasmanian Devil, Cuscus, and Quoll Birds: Tawny Frogmouth and Kiwi


Game Play:

For $5 a player has a chance to win $40. First, they will draw a token from a bag

with the picture of a celebrity face. Without replacing the token that they have

drawn, they will choose a second token

Game Pieces:

16 tokens and 1 bag. On the tokens will be pictures of…

Fozzie Barack Obama Abraham Lincoln Snow White

Kermit Nelson Mandela Stonehenge Mother Teresa

Miss Piggy Mahatma Gandhi an Oven Pop Tarts

Gonzo Chuck Norris Nitrogen Three Goats

How To Win:

If a player chooses a man followed by a Muppet they win $40. If they choose two

Muppets they will win $5. If they choose two men they will win $1. If they

choose a Muppet followed by a man, they lose. All other combinations lose.

Number of

Possible Outcomes

Probability

Distribution

Expected Value


Game Play:

For $50 a player has a chance to win $1000. First, they spin a Price is Right style

wheel with 28 equal sections. Second, they will count the number of Js or Ks in

their word.

Game Pieces:

A Wheel with 28 equal parts

Beef Pinecone Tank Castle

Kayak Taj Mahal Spaceship Joker

Church Lugubrious Squirrel Dustbuster

Mummy Kinkajou Captain Crunch Gas Station

Leonardo Boomdiggy Bulldog Nothing

Juked Franklin Kettle Lamp

Helmet Lawn Mower Ice Cube Betty Boop

How To Win:

If a player lands on a word with three Js or Ks they win $1000. If they land on a

word with two Js or Ks they win $100. If they land on a word with one J or K

they win $10. Everything else loses.

Number of

Possible Outcomes

Probability

Distribution

Expected Value


Game Room Analysis

1. Use the calculations from the games to answer the following questions.

Rock & Roll Love Connection Get in My Belly Manly Muppet JK – OMG LOL

Game Cost

Expected Value

If the Game Maker had X people play they would expect to earn _______ (**NOTE: the expected values calculated earlier were from the perspective of the player so the signs are opposite)

10

25

75

250

500

2. Which game has the highest expected value for the PLAYER? Rate the games from highest expected

value to lowest.

3. If 500 people attended the fair and played each game, which game maker would you expect to earn the

most money? How much would they make?

4. If you were the Game Maker of the Love Connection game, what would you do?

5. The owner of JK – OMG LOL is concerned that not many people are playing the game. What could be

done to fix it?

6. If you had $50 to play games, which would you choose? Why?

7. The goal of each Game Maker is to make $1,000 in a day. How many customers would they need?

Rock & Roll

Love

Connection

Get in My

Belly

Manly

Muppet

JK – OMG

LOL

Number of Customers to

earn $1,000

**You are now the GAME MAKER: You get to collect the money paid to play ANDAnything the PLAYER would lose (If the player will make money, you lose money)

1.) Use the information you gathered on the games to fill in the table.

($/person * # people)


Welcome to

Casino Royale Roll out the red carpet and pull up your sleeves because you are about to put your expected value skills to the

ultimate test. In this project, you will design a game that will ideally earn you a lot of Swagg -- Swagg Cash that

is. Each person will start by receiving exactly 125 Swagg Cash and the object of the game is to earn the most

Swagg in the class. The person will earn the title Master of Swagg.

Designing a game is more complex than it seems. It has to be appealing. It can’t be too simple that it’s boring,

but also not overly complicated. If the price is too high, people may not play. If it is too low, you may not earn

enough money. Outlining and analyzing your own game will get you ready for the Casino Royale.

Name of the Game

Game Pieces

What do you need? **You must be prepared to create or bring your game pieces to class**

Game Play

How much does it cost to play?

How does the game work?

How does the game end?

Possible Outcomes


Compute the probabilities of the outcomes, assign values and determine the expected value for your game.

Probability of Each Outcome

Assigned Value of Each Outcome

**Remember you and your customers will only have 125 Swagg Cash. Amounts of Swagg must be a combinations 1, 5, 10, 25**

Expected Value of a Game

If X people played your game how much would you earn?

X Expected Earnings

20

100

250

600

1000

5000

Do it Yourself!

Create a sign advertising your game, the cost and the rewards. To help attract customers, the sign should

briefly describe how the game is played. Each outcome should be defined with an assigned value. The sign

has to look nice or else less people will be likely to want to play your game.



Casino Royale

Player Log Sheet Before you play a game, you must calculate the expected value of playing it. You can play the same game

multiple times. If you run out of Swagg Cash, you must go see the person with the most Swagg in the room

(your teacher). The Game Designer can help you with calculations.

Game Name

Number of Possible

Outcomes

Probability

Distribution

Expected Value

Actual Result

(win/lose, gain/loss)

Game Name

Number of Possible

Outcomes

Probability

Distribution

Expected Value

Actual Result



Game Name

Number of Possible

Outcomes

Probability

Distribution

Expected Value

Actual Result


Game Name

Number of Possible

Outcomes

Probability

Distribution

Expected Value

Actual Result


Game Name

Number of Possible

Outcomes

Probability

Distribution

Expected Value

Actual Result



Casino Royale

Maker Record Sheet The Game Maker must keep a record of all transactions and the results of each game in the table.

Game Name:

Player’s Name Result


Player’s Name Result


Casino Royale Reflections

Complete the reflection questions from the two different perspectives in the project.

Game Maker Game Player

1a. How much Swagg Cash did you end with? Did

you gain or lose?

1b. How much Swagg Cash did you end with? Did

you gain or lose?

2a. As a Maker, were you successful? Describe how

things went. What could you have done differently?

2b. As a Player, were you successful? Describe how

things went. What could you have done differently?

3a. How did the expected value of your game

compare to the other games? Do you think it affected

your business?

3b. Which games had the highest expected value?

Which games did you earn the most Swagg Cash?

4a. How did your expected value hold up? Multiply

the calculated expected value by the number of

customers you had. Did you gain or lose a similar

amount? Why or why not?

4b. Did games with better expected values hold up?

Did you perform better in games with higher values?

Did you perform poorly in games with lower values?

5. Did you perform better as a Maker or a Player? Which did you do first? Did you have a strategy? Did your

perspective in your first role affect you in your second?

Total Swagg Cash:

(Game Maker Total + Game Player Total)

______________________________________________

Documents

Expectations - Ms. Hogan's Classes · 2018. 2. 26. · How To Win: If a player chooses a man followed by a Muppet they win $40. If they choose two Muppets they will win $5. If they